Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter The Normal Probability Distribution 7

Chap 22

3 Jury Selection Problem There are 60 women and 40 men waiting outside the courtroom in the jury pool. There are 60 women and 40 men waiting outside the courtroom in the jury pool. Assuming the probability of being selected for the jury is the same for each person, what is the probability that the 12 person jury selected will consist of 7 men and 5 women? Assuming the probability of being selected for the jury is the same for each person, what is the probability that the 12 person jury selected will consist of 7 men and 5 women?

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Properties of the Normal Distribution 7.1

A probability density function (pdf) is an equation used to compute probabilities of continuous random variables. The total area under the graph of the equation over all possible values of the random variable must equal 1. The area under the graph of the density function over an interval represents the probability of observing a value of the random variable in that interval. A probability density function (pdf) is an equation used to compute probabilities of continuous random variables. The total area under the graph of the equation over all possible values of the random variable must equal 1. The area under the graph of the density function over an interval represents the probability of observing a value of the random variable in that interval. 5-5

7-6 “Probability Density Function” is the equation of the blue curve. This equation gives the “y” values of the curve as “x” moves from left to right. The “area under the curve” is the area between the curve (blue line) and the horizontal.

7-7 Relative frequency histograms that are symmetric /bell-shaped are said to have the shape of a normal curve.

7-8 If a continuous random variable is “normally distributed”, or has a “normal probability distribution”, then a rel freq histogram of that variable has the shape of a “normal” or symmetric bell-shaped curve

Properties of the Normal Curve 1. It is symmetric about its mean, μ. 2. Mean = median = mode. The curve has a single maximum point at x = μ. 3. It has inflection points at μ ± σ 4. The area under the curve is The area under the curve to the right of μ, as well as to the left of μ, equals 1/2. Properties of the Normal Curve 1. It is symmetric about its mean, μ. 2. Mean = median = mode. The curve has a single maximum point at x = μ. 3. It has inflection points at μ ± σ 4. The area under the curve is The area under the curve to the right of μ, as well as to the left of μ, equals 1/2. 5-9

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6.As “x” increases without bound (gets further from the mean), the graph approaches, but never reaches, the horizontal axis. 7.The Empirical Rule ( ): Approximately 68% of the area under the normal curve lies between μ ± 1 σ; approx. 95% of the area lies between μ ± 2σ; and approx. 99.7% of the area lies between μ ± 3σ. 6.As “x” increases without bound (gets further from the mean), the graph approaches, but never reaches, the horizontal axis. 7.The Empirical Rule ( ): Approximately 68% of the area under the normal curve lies between μ ± 1 σ; approx. 95% of the area lies between μ ± 2σ; and approx. 99.7% of the area lies between μ ± 3σ. 5-11

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7-13 The data on the next slide represent the heights (in) of a random sample of 50 two-year old males. (a) Draw a histogram of the data using first class boundaries of 31.5 and 32.5 in. (b) Do you think that the variable: “x” = height of 2-year old males is normally distributed? A Normal Continuous Random Variable

HEIGHT (IN) OF TWO-YR OLD MALES

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7-16 In the next slide, we have a normal density curve superimposed over the histogram. How does the area of the rectangle corresponding to a height between 34.5 and 35.5 inches relate to the area under the curve between these two heights?

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Area under a Normal Curve Suppose that a random variable “x” is normally distributed with mean μ and standard deviation σ. The area under the normal curve for any interval of values of “x” (say, 34.5 – 35.5 in) represents either: 1.the proportion of the population with that interval (range) of values or: 2. the probability that a randomly selected individual from the population will have that range of values. Area under a Normal Curve Suppose that a random variable “x” is normally distributed with mean μ and standard deviation σ. The area under the normal curve for any interval of values of “x” (say, 34.5 – 35.5 in) represents either: 1.the proportion of the population with that interval (range) of values or: 2. the probability that a randomly selected individual from the population will have that range of values. 5-18

7-19 EXAMPLE: Area Under a Normal Curve Suppose, the weights of adult African giraffes are normally distributed with mean μ = 2200 pounds and standard deviation σ = 200 pounds. (a) Shade the area under the normal curve to the left of x = 2100 pounds. (b) Suppose that the area under the normal curve to the left of x = 2100 pounds is Provide two (Proportion/Probability) interpretations of this result.

7-20 EXAMPLEArea Under a Normal Curve Adult Giraffe: μ = 2200 pounds and σ = 200 pounds 1)The proportion of giraffes whose weight is less than 2100 pounds is or 30.85% of the giraffe population. 2) The probability that a randomly selected giraffe weighs less than 2100 pounds is

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Applications of the Normal Distribution 7.2

Standardizing a Normal Random Variable Suppose that the random variable X is normally distributed with mean μ and standard deviation σ. Then the random variable is normally distributed with mean μ = 0 and standard deviation σ = 1.The random variable Z has a “standard” normal distribution. Standardizing a Normal Random Variable Suppose that the random variable X is normally distributed with mean μ and standard deviation σ. Then the random variable is normally distributed with mean μ = 0 and standard deviation σ = 1.The random variable Z has a “standard” normal distribution. 5-22

7-23 “Standard” Normal Distribution

7-24 IQ scores can be modeled by a normal distribution with μ = 100 and σ = 15. An IQ score of 120, is 1.33 standard deviations above the mean.

25 Normal Probability Accuracy For these probabilities, we require 4-decimal accuracy, so set your calculator to: For these probabilities, we require 4-decimal accuracy, so set your calculator to: Mode: Float: 4 Mode: Float: 4

7-26 Table IV: the area under the standard normal curve to the left of z = 1.33 is

7-27 Find the area to the right of z = 1.33.

7-28 Areas Under the Standard Normal Curve Empirical Rule:

7-29 Find the area under the standard normal curve to the left of z = – Area to the left of z = – 0.38 is

7-30 Find the area under the standard normal curve to the right of z = 1.25 Area right of 1.25 = 1 – area left of 1.25 = 1 – =

7-31 Find the area under the standard normal curve between z = –1.02 and z = 2.94 Area between z = –1.02 and 2.94 = (Area left of z = 2.94) – (area left of z = –1.02) = – =

TI-84 2:Distr: normcdf… (lo, up, mn, stddev) (using z-score): normcdf (-1,1) = normcdf (-3.49,-0.38) = normcdf (-1E99,-0.38) = normcdf (-3.49,-0.38) = normcdf (-1E99,-0.38) =

TI-84 2:Distr: normcdf… (lo, up, mn, stddev) If you wanted to know the probability of getting 120 or less on an IQ test whose mean is 100 and stddev is 15… If you wanted to know the probability of getting 120 or less on an IQ test whose mean is 100 and stddev is 15… (using x-score) normcdf (0,120,100,15) = (using x-score) normcdf (0,120,100,15) = (using z-score): normcdf (-1E99,1.333) = (using z-score): normcdf (-1E99,1.333) = (using lo bd of z= gives ) (using lo bd of z= gives )

invNorm (area/prob, mean, stddev) Find the raw score/z-score that corresponds to on the standard IQ Test (100;15 ) (using x-score) InvNorm (0.85,100,15) = (using x-score) InvNorm (0.85,100,15) = (using z-score) InvNorm (0.85) = (using z-score) InvNorm (0.85) =

TI-84 sample problems… normcdf (-1.5,1.25) → normcdf (-1.5,1.25) → normcdf (0,133,100,15) → normcdf (0,133,100,15) → (89,133,100,15) → (89,133,100,15) → invNorm (0.9861) → invNorm (0.9861) → invNorm (0.9861,100,15) → invNorm (0.9861,100,15) → normcdf (-1E99,1.25) → normcdf (-1E99,1.25) →

7-36 The combined (verbal + quantitative) score on the GRE (Graduate Record Exam (GRE ~ to get accepted into graduate school) is normally distributed with mean=1049 and stddev =189 Finding the Value of a Normal Variable What is the raw test score for a student whose rank is at the 85 th percentile?

7-37 Value of a Normal Random Variable The z-score that corresponds to the 85 th percentile is the z-score which has.8500 area to its left. TI-84 or Table V : InvNorm (0.8500) = So, this z-score is x = µ + zσ = (189) = or 1245 A person who scores 1245 on the GRE would rank in the 85 th percentile, or the prob a person scores 1245 on the GRE is 0.850

7-38 Karlos, Inc. manufactures construction-grade steel rebar rods. Their length is normally distributed with a mean of 100 cm (1 m) and a standard deviation of 0.45 cm. Suppose Quality Control wants to accept 90% of all rods manufactured, but rebuild all rods outside that tolerance. Determine the min/max length of these rods that make up the middle 90% of all rods manufactured. Value of a Normal Random Variable

7-39 Value of a Normal Random Variable Area = 0.05 InvNorm(0.05) = z 1 = –1.645 and z 2 = x 1 = µ + z 1 σ = (–1.645)(0.45) = cm OR InvNorm(0.05,100,0.45) = = 99.3 cm The steel rods that make up the middle 90% would have lengths between 99.3 cm and cm.

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Assessing Normality (Skip) 7.3

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section The Normal Approximation to the Binomial Probability Distribution 7.4

Criteria for a Binomial Probability Experiment An experiment is said to be a binomial experiment if: 1. The experiment is performed “n” independent times. Each repetition is called a trial. Independence means that the outcome of one trial will not affect the outcome of the other trials. 2. For each trial, there are only two possible disjoint outcomes: Success / Failure (T/F; Yes/No, H/T, etc.) 3. The probability of success, “p” and the prob of failure “q”, are the same for each trial of the experiment. Note: p + q =1 and q = 1- p. Criteria for a Binomial Probability Experiment An experiment is said to be a binomial experiment if: 1. The experiment is performed “n” independent times. Each repetition is called a trial. Independence means that the outcome of one trial will not affect the outcome of the other trials. 2. For each trial, there are only two possible disjoint outcomes: Success / Failure (T/F; Yes/No, H/T, etc.) 3. The probability of success, “p” and the prob of failure “q”, are the same for each trial of the experiment. Note: p + q =1 and q = 1- p. 5-42

7-43 For a fixed prob of success “p”, as the number of trials “n” in a binomial experiment increases, the probability distribution of “x” becomes more nearly symmetric and bell-shaped. As a rule of thumb, if the variance = npq > 10, then the probability distribution will be approx normal (bell-shaped) and we can use “z” scores to predict probability of experiment outcomes.

The Normal Approximation to the Binomial Probability Distribution If npq ≥ 10, (stddev = ≥ 3.2), the binomial random variable “x” is approximately normally distributed with : Mean: μ X = np Std Dev: Note: “q” = (1- p) The Normal Approximation to the Binomial Probability Distribution If npq ≥ 10, (stddev = ≥ 3.2), the binomial random variable “x” is approximately normally distributed with : Mean: μ X = np Std Dev: Note: “q” = (1- p) 5-44

7-45 P(X = 18) ≈ P(17.5 < X < 18.5)

7-46 P(X < 18) ≈ P(X < 18.5)

7-47 Exact Probability Using Binomial: P(X ≤ a) Approximate Probability Using Normal: P(X ≤ a + 0.5)

7-48 Exact Probability Using Binomial: P(X ≥ a) Approximate Probability Using Normal: P(X ≥ a – 0.5)

7-49 Exact Probability Using Binomial: P(a ≤ X ≤ b) Approximate Probability Using Normal: P(a=0.5 ≤ X ≤ b+0.5)

7-50 Using the Binomial Probability Distribution Function 35% of all car-owning households have three or more cars. (a)In a sample of 400 car-owning households, what is the probability that fewer than 150 have three or more cars? P(x<150) ≈ P(x<150.5)

7-51 Using the Binomial Probability Distribution Function 35% of all car-owning households have three or more cars. (b) In a sample of 400 car-owning households, what is the probability that at least 160 have three or more cars?

Chap 252